Tensor Decomposition with Missing Indices

We can denote $N \in \mathbb{N}$ observations, by $X=\{x_n\}_{n=1}^{N}$ the values and by $\hat{Z}=\{(\hat{i}_n, \hat{j}_n, \hat{k}_n)\}_{n=1}^{N}$ the corresponded potential missing indices;
and the missing index can be denoted as $\emptyset$.